International Monetary Fund managing director Christine Lagarde was optimistic about the deal on Thursday.

"As we speak, it looks like it's going through, and it looks as if the numbers will be promising," she told PBS television.

Shares rose on optimism that the deal had been supported, with both the German and French share indexes earlier ending up 2.5%, while the UK's FTSE 100 added 1.2%.

In New York, the main Dow Jones index closed up 0.6%.

Bailout requirement

Greece had said it wanted 90% of bondholders, such as banks and pension funds, to agree to take a 53.5% cut in the 206bn euros ($272bn; £172bn) of Greek bonds they hold. But it only requires a 75% take-up to be able to force through the deal.

German reinsurance group Munich Re, French banks Societe Generale and BNP Paribas, and some pension funds, had said they would sign up.

But some small pension funds had said they would not back the swap, and others said they were waiting to see what hedge funds will do.

One bondholder told the BBC that he had "no incentive" to accept the deal and would not do so.

"I'm not in the business for altruistic reasons," said Patrick Armstrong, managing partner at Armstrong Investment Managers. "Capital markets function best when people are out to deliver return on capital investment."

'Full participation'

The European Union and International Monetary Fund have said that if the debt swap does not go through then Greece will not get its latest bailout of 130bn euros.

Analysis

By Mark LowenBBC News, Athens

When it comes to Greece's finances, it's rare that targets are met and things go according to plan.

But this bond deal looks like it's going that way.

Reports suggest Greece is on course to achieve the participation rate it needs for the exchange to pass with fewer problems than expected.

Over 75% of bondholders are said to have signed up; the prime minister said he expected "maximum participation".

The brinkmanship strategy from the government seems to have worked. Greece's Finance Minister warned that bondholders waiting for a better deal would be forced into the swap and would simply not receive the value of their original bonds.

If the deal goes through, there will be a new glint of hope here: a sense that Greece would be in a better position to begin to manage its debt burden.

A small but significant step forward for this recession-stricken country.

Economic and Monetary Affairs Commissioner Olli Rehn said earlier this week that there would be no better offer, and the deal was vital for eurozone financial stability.

He said: "It is important that all investors recognise that Europe has committed the maximum funds available to this voluntary debt exchange and that full participation is necessary for the Greek programme to move forward."

The Greek Finance Ministry had made it clear that the alternative to the debt swap is a potential default.

"The republic's representative noted that if [private sector involvement] is not successfully completed, the official sector will not finance Greece's economic programme and Greece will need to restructure its debt," it said on Tuesday.

Athens was first bailed out in 2010 with 109bn euros from the EU and IMF.

As the sovereign debt woes in the eurozone continue to focus primarily on Greece, the head of the World Trade Organization, Pascal Lamy, told the BBC that eurozone nations would have to align their economies more closely.

He said: "What the Europeans have to do is align the level of integration, of solidarity, which they have currency wise, monetary wise, with something which is economically much more convergent.